Explain why or why not Determine whether the following statements are true and give an explanation or counterexample


d. According to Newton’s Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

Verified step by step guidance

1. Recall Newton's Law of Cooling, which states that the rate of change of the temperature \(T(t)\) of an object is proportional to the difference between its temperature and the ambient temperature \(T_a\). Mathematically, this is expressed as: \[\frac{dT}{dt} = -k (T - T_a)\] where \(k > 0\) is a constant depending on the characteristics of the object and environment.
2. Solve this first-order differential equation by separating variables or recognizing it as a linear ODE. The general solution is: \[T(t) = T_a + (T_0 - T_a) e^{-k t}\] where \(T_0\) is the initial temperature of the object at time \(t=0\).
3. Analyze the behavior of \(T(t)\) as \(t \to \infty\). Since \(e^{-k t}\) approaches zero as \(t\) becomes very large, the temperature \(T(t)\) approaches the ambient temperature \(T_a\) asymptotically.
4. Understand the meaning of "reaching" the ambient temperature. Because the exponential term never becomes exactly zero for any finite \(t\), the temperature \(T(t)\) never equals \(T_a\) at any finite time, but only approaches it infinitely closely as time goes to infinity.
5. Conclude that according to Newton's Law of Cooling, the temperature of a hot object does not reach the ambient temperature after a finite amount of time, but rather approaches it asymptotically. Therefore, the statement is false.

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature. This results in an exponential decay of the temperature difference over time, describing how objects cool or warm toward the surrounding temperature.

Exponential Decay and Limits

Exponential decay functions approach a limiting value asymptotically, meaning they get closer and closer to a certain temperature but never actually reach it in finite time. This concept explains why the temperature difference decreases continuously but does not become zero at any finite moment.

Finite Time vs. Infinite Time in Cooling Processes

In cooling processes governed by Newton's Law, the temperature difference approaches zero only as time approaches infinity. Therefore, the object never exactly reaches ambient temperature in finite time, highlighting the difference between practical approximation and mathematical exactness.